While we can't divide by zero, we can still meaningfully ask
what happens to a ratio $\frac{f(x)}{g(x)}$ when both $f(x)$ and
$g(x)$ go to zero. In fact, this sort of ratio appears every
time we take a derivative: $$ f'(x) = \lim_{h \to 0}
\frac{f(x+h)-f(x)}{h}.$$ We can't just plug in $h=0$, since then
both the numerator and denominator would be zero.
Derivatives can help us evaluate limits that look like
$\frac{0}{0}$.

L'Hospital's
Rule:
If $f(x)$ and $g(x)$ are continuous functions with either
$f(a)=g(a)=0$, or $f(a)=\pm\infty$ and $g(a)=\pm\infty$,
then $$\lim_{x \to a}\, \frac{f(x)}{g(x)} = \lim_{x \to a}\,
\frac{f'(x)}{g'(x)}.$$

Warning: While
L'Hospital's Rule says that the limits
of $\frac{f}{g}$ and $\frac{f'}{g'}$ are equal under certain
circumstances, it is not true that $\frac{f}{g}=\frac{f'}{g'}$!

In addition, limits that look like $0 \cdot \infty$, $\infty -
\infty$, $1^\infty$, $\infty^0$, or $0^0$ can be manipulated to
use L'Hospital's rule. All these limits are called indeterminate forms, which means
that the limit value may be $0,\infty,-\infty,54,-100$ or
anything. We can tell nothing without doing more work.
We have seen some indeterminate forms in the past, and now we have
another tool to use when we do more work.

Let us look at these ideas, and see intuitively why L'Hospital's
rule works.